The G0 polarimetric distribution

Density and random generation for the polarimetric $\mathcal{G}^0_m(\alpha, \Sigma, L)$ law (also known as $\mathcal{G}^0_P$), obtained from the multiplicative model $Z = X Y$ with $Y \sim \mathcal{W}_m(\Sigma, L)$ and reciprocal gamma texture $X \sim \Gamma^{-1}(-\alpha, -\alpha - 1)$ with unit mean (Freitas et al., 2005): $$f(Z) = \frac{L^{mL}\Gamma(mL - \alpha)|Z|^{L-m}} {|\Sigma|^L \Gamma_m(L) \Gamma(-\alpha)(-\alpha-1)^{\alpha}} \left[L\,\mathrm{tr}(\Sigma^{-1}Z) - \alpha - 1\right]^{\alpha - mL},$$ with $\alpha < -1$. When $\alpha \to -\infty$ it converges to the scaled complex Wishart law.

Usage

dg0p(x, alpha, Sigma, L, log = FALSE)

rg0p(n, alpha, Sigma, L, output = c("array", "vec"))

Arguments

x: PolSAR observations: a complex array m x m x n, a single Hermitian matrix or an n x m(m+1)/2 matrix (see as_polsar_vec()).
alpha: Roughness parameter ($\alpha < -1$).
Sigma: Hermitian positive definite m x m matrix (the mean).
L: Number of looks ($L > m - 1$).
log: Logical; if TRUE the log-density is returned.
n: Number of observations.
output: Format of the generated sample: "array" (m x m x n) or "vec" (n x m(m+1)/2).

Value

dg0p() returns (log-)densities and rg0p() the simulated sample.

References

Freitas, C. C., Frery, A. C. and Correia, A. H. (2005). The polarimetric G distribution for SAR data analysis. Environmetrics, 16(1), 13–31.

Nascimento, A. D. C., Ferreira, J. A. and Frery, A. C. (2023). Unsupervised segmentation of PolSAR data with complex Wishart and G0m distributions and Shannon entropy. Statistics and Computing, 33, 129. doi:10.1007/s11222-023-10298-0

Examples

Sigma <- polsar_example_sigma("urban")
set.seed(1)
Z <- rg0p(3, alpha = -3, Sigma = Sigma, L = 4)
dg0p(Z, alpha = -3, Sigma = Sigma, L = 4, log = TRUE)
#> [1]  7.655394 12.416744  6.848470